Why is Δ_t smaller than Δ_o in the 4/9 rule?

Fewer ligands (four vs six) and different angular overlap factors in the crystal-field model reduce the d-orbital splitting in tetrahedral symmetry relative to octahedral for comparable metal–ligand distances.

Does this include Jahn–Teller distortion or π-backbonding?

No. Those effects change orbital energies and gaps; this is a two-level crystal-field toy with a single adjustable Δ and a scalar P.

Is the color swatch quantitative?

No. It is a monotonic hue vs Δ_oct teaching aid. Experimental colors depend on ligands, oxidation state, geometry, and many overlapping transitions.

Crystal Field Splitting (Oct / Tet)

Δ_t = (4/9)Δ_o; HS vs LS d⁴–d⁷ from CFSE + pairing P; toy complex hue vs Δ_oct.

Spectrochemical hint (toy)

Strong field (larger Δ) → shorter toy λ_d–d ~833 nm

d-orbital splitting

d_z²+0.6·Δ

d_x²−y²+0.6·Δ

Δ = 12000 cm⁻¹

d_xy-0.4·Δ

d_xz-0.4·Δ

d_yz-0.4·Δ

t₂g lower · e_g upper

HS vs LS energy (same n)

E_HS = 13200 cm⁻¹ · E_LS = 25200 cm⁻¹

HS lower — fewer pairs despite occupying upper orbitals.

d⁴–d⁷ in O_h are the usual HS/LS crossover cases; T_d often stays HS because Δ_t is smaller.

About this model

In ligand field theory, an octahedral (O_h) environment splits the five degenerate d orbitals into a lower t₂g triplet and a higher e_g doublet with gap Δ_o; a tetrahedral (T_d) field inverts the pattern into a lower e doublet and upper t₂ triplet with gap Δ_t. For the same electrostatic point-charge model, Δ_tet = (4/9) Δ_oct. This page uses the standard relative one-electron energies (t₂g at −0.4Δ_o, e_g at +0.6Δ_o; e at −0.6Δ_t, t₂ at +0.4Δ_t) and adds a phenomenological pairing energy P per spin pair. High-spin (Hund) filling maximizes unpaired electrons before pairing within the energy-ordered set; low spin fills the lower subshell completely before occupying the upper set. For d⁴–d⁷ in octahedral symmetry, comparing E = CFSE + (#pairs)·P predicts the HS/LS crossover when Δ and P are varied. Tetrahedral gaps are smaller in the same model, so HS is often favored. The colored swatch maps Δ_oct to a toy hue for spectrochemical intuition only; real colors come from multiplet transitions, selection rules, and charge-transfer bands.

Who it's for: Inorganic chemistry after d-block introduction; pairs with orbital-shapes and electron-config pages.

Key terms

crystal field splitting
Δ_o
Δ_t
pairing energy
high spin
low spin
CFSE
spectrochemical series

How it works

Octahedral (O_h) and tetrahedral (T_d) crystal fields split the five d orbitals into two sets with gap Δ. For the same ligand–metal model, Δ_tet = (4/9) Δ_oct. High spin (HS) vs low spin (LS) for d⁴–d⁷ compares pairing energy P with Δ: larger Δ (strong field / π-acceptor ligands) favors LS when it lowers CFSE + pairing. Color here is a toy map from Δ (spectrochemical intuition), not a ligand-field multiplet calculation.

Key equations

O_h: ε(t₂g) = −0.4Δ_o , ε(e_g) = +0.6Δ_o · T_d: ε(e) = −0.6Δ_t , ε(t₂) = +0.4Δ_t

Δ_t = (4/9) Δ_o (same point-charge model scaling)

Frequently asked questions

Why is Δ_t smaller than Δ_o in the 4/9 rule?: Fewer ligands (four vs six) and different angular overlap factors in the crystal-field model reduce the d-orbital splitting in tetrahedral symmetry relative to octahedral for comparable metal–ligand distances.
Does this include Jahn–Teller distortion or π-backbonding?: No. Those effects change orbital energies and gaps; this is a two-level crystal-field toy with a single adjustable Δ and a scalar P.
Is the color swatch quantitative?: No. It is a monotonic hue vs Δ_oct teaching aid. Experimental colors depend on ligands, oxidation state, geometry, and many overlapping transitions.

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